high school contest problems

USAAMC 10

2010

A

1 Mary’s top book shelf holds five books with the following widths, in centimeters: 6, 12 , 1, 2.5,

and 10. What is the average book width, in centimeters?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

2 Four identical squares and one rectangle are placed together to form one large square asshown. The length of the rectangle is how many times as large as its width?

(A) 54 (B) 4

3 (C) 32 (D) 2 (E) 3

3 Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marblesot Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles didTyrone give to Eric?

(A) 3 (B) 13 (C) 18 (D) 25 (E) 29

4 A book that is to be recorded onto compact discs takes 412 minutes to read aloud. Each disccan hold up to 56 minutes of reading. Assume that the smallest possible number of discs isused and that each disc contains the same length of reading. How many minutes of readingwill each disc contain?

(A) 50.2 (B) 51.5 (C) 52.4 (D) 53.8 (E) 55.2

5 The area of a circle whose circumference is 24! is k!. What is the value of k?

(A) 6 (B) 12 (C) 24 (D) 36 (E) 144

6 For positive numbers x and y the operation !(x, y) is defined as

!(x, y) = x" 1y

http://www.artofproblemsolving.com/This file was downloaded from the AoPS Math Olympiad Resources Page

http://www.artofproblemsolving.com/

http://www.artofproblemsolving.com/Forum/resources.php?c=182

http://www.mathlinks.ro/Forum/resources.php?c=182&cid=43

http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43&year=2010

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1770740






USAAMC 10

2010

What is !(2,!(2, 2))?

(A) 23 (B) 1 (C) 4

3 (D) 53 (E) 2

7 Crystal has a running course marked out for her daily run. She starts this run by headingdue north for one mile. She then runs northeast for one mile, then southeast for one mile.The last portion of her run takes her on a straight line back to where she started. How far,in miles is this last portion of her run?

(A) 1 (B)#

2 (C)#

3 (D) 2 (E) 2#

2

8 Tony works 2 hours a day and is paid $0.50 per hour for each full year of his age. During asix month period Tony worked 50 days and earned $630. How old was Tony at the end of thesix month period?

(A) 9 (B) 11 (C) 12 (D) 13 (E) 14

9 A palindrome, such as 83438, is a number that remains the same when its digits are reversed.The numbers x and x + 32 are three-digit and four-digit palindromes, respectively. What isthe sum of the digits of x?

(A) 20 (B) 21 (C) 22 (D) 23 (E) 24

10 Marvin had a birthday on Tuesday, May 27 in the leap year 2008. In what year will hisbirthday next fall on a Saturday?

(A) 2011 (B) 2012 (C) 2013 (D) 2015 (E) 2017

11 The length of the interval of solutions of the inequality a $ 2x + 3 $ b is 10. What is b" a?

(A) 6 (B) 10 (C) 15 (D) 20 (E) 30

12 Logan is constructing a scaled model of his town. The city’s water tower stands 40 metershigh, and the top portion is a sphere that holes 100, 000 liters of water. Logan’s miniaturewater tower holds 0.1 liters. How tall, in meters, should Logan make his tower?

(A) 0.04 (B) 0.4! (C) 0.4 (D) 4

! (E) 4

13 Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After thestop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total triptime of 3 hours including the stop. Which equation could be used to solve for the time t inhours that she drove before her stop?

(A) 80t + 100(8/3" t) = 250 (B) 80t = 250 (C) 100t = 250(D) 90t = 250 (E) 80(8/3" t) + 100t = 250













USAAMC 10

2010

14 Triangle ABC has AB = 2 · AC. Let D and E be on AB and BC, respectively, such that!BAE = !ACD. Let F be the intersection of segments AE and CD, and suppose that%CFE is equilateral. What is !ACB?

(A) 60! (B) 75! (C) 90! (D) 105! (E) 120!

15 In a magical swamp there are two species of talking amphibians: toads, whose statements arealways true, and frogs, whose statements are always false. Four amphibians, Brian, Chris,LeRoy, and Mike live together in the swamp, and they make the following statements:

Brian: ”Mike and I are di!erent species.” Chris: ”LeRoy is a frog.” LeRoy: ”Chris is a frog.”Mike: ”Of the four of us, at least two are toads.”

How many of these amphibians are frogs?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

16 Nondegenerate %ABC has integer side lengths, BD is an angle bisector, AD = 3, andDC = 8. What is the smallest possible value of the perimeter?

(A) 30 (B) 33 (C) 35 (D) 36 (E) 37

17 A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center ofeach face. The edges of each cut are parallel to the edges of the cube, and each hole goes allthe way through the cube. What is the volume, in cubic inches, of the remaining solid?

(A) 7 (B) 8 (C) 10 (D) 12 (E) 15

18 Bernardo randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and arrangesthem in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbersfrom the set {1, 2, 3, 4, 5, 6, 7, 8} and also arranges them in descending order to form a 3-digitnumber. What is the probability that Bernardo’s number is larger than Silvia’s number?

(A) 4772 (B) 37

56 (C) 23 (D) 49

72 (E) 3956

19 Equiangular hexagon ABCDEF has side lengths AB = CD = EF = 1 and BC = DE =FA = r. The area of %ACE is 70

(A) 4"

33 (B) 10

3 (C) 4 (D) 174 (E) 6

20 A fly trapped inside a cubical box with side length 1 meter decides to relieve its boredom byvisiting each corner of the box. It will begin and end in the same corner and visit each ofthe other corners exactly once. To get from a corner to any other corner, it will either fly orcrawl in a straight line. What is the maximum possible length, in meters, of its path?

(A) 4 + 4#

2 (B) 2 + 4#

2 + 2#

3 (C) 2 + 3#

2 + 3#

3 (D) 4#

2 + 4#

3(E) 3

#2 + 5

#3













USAAMC 10

2010

21 The polynomial x3 " ax2 + bx " 2010 has three positive integer zeros. What is the smallestpossible value of a?

(A) 78 (B) 88 (C) 98 (D) 108 (E) 118

22 Eight points are chosen on a circle, and chords are drawn connecting every pair of points. Nothree chords intersect in a single point inside the circle. How many triangles with all threevertices in the interior of the circle are created?

(A) 28 (B) 56 (C) 70 (D) 84 (E) 140

23 Each of 2010 boxes in a line contains a single red marble, and for 1 $ k $ 2010, the box in thekth position also contains k white marbles. Isabella begins at the first box and successivelydraws a single marble at random from each box, in order. She stops when she first draws ared marble. Let P (n) be the probability that Isabella stops after drawing exactly n marbles.What is the smallest value of n for which P (n) < 1

2010?

(A) 45 (B) 63 (C) 64 (D) 201 (E) 1005

24 The number obtained from the last two nonzero digits of 90! is equal to n. What is n?

(A) 12 (B) 32 (C) 48 (D) 52 (E) 68

25 Jim starts with a positive integer n and creates a sequence of numbers. Each successivenumber is obtained by subtracting the largest possible integer square less than or equal tothe current number until zero is reached. For example, if Jim starts with n = 55, then hissequence contains 5 numbers:

55

55" 72 = 6

6" 22 = 2

2" 12 = 1

1" 12 = 0

Let N be the smallest number for which Jim’s sequence has 8 numbers. What is the unitsdigit of N?

(A) 1 (B) 3 (C) 5 (D) 7 (E) 9











USAAMC 10

2010

B

1 What is 100(100" 3)" (100 · 100" 3)?

(A) " 20, 000 (B) " 10, 000 (C) " 297 (D) " 6 (E) 0

2 Makayla attended two meetings during her 9-hour work day. The first meeting took 45minutes and the second meeting took twice as long. What percent of her work day was spentattending meetings?

(A) 15 (B) 20 (C) 25 (D) 30 (E) 35

3 A drawer contains red, green, blue, and white socks with at least 2 of each color. What isthe minimum number of socks that must be pulled from the drawer to guarantee a matchingpair?

(A) 3 (B) 4 (C) 5 (D) 8 (E) 9

4 For a real number x, define &(x) to be the average of x and x2. What is &(1)+&(2)+&(3)?

(A) 3 (B) 6 (C) 10 (D) 12 (E) 20

[Thanks PowerOfPi, that’s exactly how the heart looks like.]

5 A month with 31 days has the same number of Mondays and Wednesdays. How many of theseven days of the week could be the first day of this month?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

6 A circle is centered at O, AB is a diameter and C is a point on the circle with !COB = 50!.What is the degree measure of !CAB?

(A) 20 (B) 25 (C) 45 (D) 50 (E) 65

7 A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to thearea of the rectangle. What is the perimeter of this rectangle?

(A) 16 (B) 24 (C) 28 (D) 32 (E) 36

8 A ticket to a school play costs x dollars, where x is a whole number. A group of 9th gradersbuys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of$64. How many values of x are possible?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5














USAAMC 10

2010

9 Lucky Larry’s teacher asked him to substitute numbers for a, b, c, d, and e in the expressiona " (b " (c " (d + e))) and evaluate the result. Larry ignored the parentheses but addedand subtracted correctly and obtained the correct result by coincedence. The numbers Larrysubstituted for a, b, c, and d were 1, 2, 3, and 4, respectively. What number did Larrysubstitute for e?

(A) " 5 (B) " 3 (C) 0 (D) 3 (E) 5

10 Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 milesper hour if it is raining. Today she drove in the sun in the morning and in the rain in theevening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?

(A) 18 (B) 21 (C) 24 (D) 27 (E) 30

11 A shopper plans to purchase an item that has a listed price greater than $100 and can useany one of the three coupns. Coupon A gives 15% o! the listed price, Coupon B gives $30the listed price, and Coupon C gives 25% o! the amount by which the listed price exceeds$100.

Let x and y be the smallest and largest prices, respectively, for which Coupon A saves atleast as many dollars as Coupon B or C. What is y " x?

(A) 50 (B) 60 (C) 75 (D) 80 (E) 100

12 At the beginning of the school year, 50% of all students in Mr. Well’s math class answered”Yes” to the question ”Do you love math”, and 50% answered ”No.” At the end of the schoolyear, 70% answered ”Yes” and 30% answered ”No.” Altogether, x% of the students gave adi!erent answer at the beginning and end of the school year. What is the di!erence betweenthe maximum and the minimum possible values of x?

(A) 0 (B) 20 (C) 40 (D) 60 (E) 80

13 What is the sum of all the solutions of x = |2x" |60" 2x ' ?

(A) 32 (B) 60 (C) 92 (D) 120 (E) 124

14 The average of the numbers 1, 2, 3, ..., 98, 99, and x is 100x. What is x?

(A) 49101 (B) 50

101 (C) 12 (D) 51

101 (E) 5099

15 On a 50-question multiple choice math contest, students receive 4 points for a correct answer,0 points for an answer left blank, and -1 point for an incorrect answer. Jesse’s total scoreon the contest was 99. What is the maximum number of questions that Jesse could haveanswered correctly?

(A) 25 (B) 27 (C) 29 (D) 31 (E) 33













USAAMC 10

2010

16 A square of side length 1 and a circle of radius#

3/3 share the same center. What is the areainside the circle, but outside the square?

(A) !3 " 1 (B) 2!

9 ""

33 (C) !

18 (D) 14 (E) 2!/9

17 Every high school in the city of Euclid sent a team of 3 students to a math contest. Eachparticipant in the contest received a di!erent score. Andrea’s score was the median among allstudents, and hers was the highest score on her team. Andrea’s teammates Beth and Carlaplaced 37th and 64th, respectively. How many schools are in the city?

(A) 22 (B) 23 (C) 24 (D) 25 (E) 26

18 Positive integers a, b, and c are randomly and independently selected with replacement fromthe set {1, 2, 3, . . . , 2010}. What is the probability that abc + ab + a is divisible by 3?

(A)13

(B)2981

(C)3181

(D)1127

(E)1327

19 A circle with center O has area 156!. Triangle ABC is equilateral, BC is a chord on thecircle, OA = 4

#3, and point O is outside %ABC. What is the side length of %ABC?

(A) 2#

3 (B) 6 (C) 4#

3 (D) 12 (E) 18

20 Two circles lie outside regular hexagon ABCDEF . The first is tangent to AB, and the secondis tangent to DE. Both are tangent to lines BC and FA. What is the ratio of the area ofthe second circle to that of the first circle?

(A) 18 (B) 27 (C) 36 (D) 81 (E) 108

21 A palindrome between 1000 and 10, 000 is chosen at random. What is the probability that itis divisible by 7?

(A)110

(B)19

(C)17

(D)16

(E)15

22 Seven distinct pieces of candy are to be distributed among three bags. The red bag and theblue bag must each receive at least one piece of candy; the white bag may remain empty.How many arrangements are possible?

(A) 1930 (B) 1931 (C) 1932 (D) 1933 (E) 1934

23 The entries in a 3 ( 3 array include all the digits from 1 through 9, arranged so that theentries in every row and column are in increasing order. How many such arrays are there?

(A) 18 (B) 24 (C) 36 (D) 42 (E) 60

24 A high school basketball game between the Raiders and Wildcats was tied at the end of thefirst quarter. The number of points scored by the Raiders in each of the four quarters formed















USAAMC 10

2010

an increasing geometric sequence, and the number of points scored by the Wildcats in each ofthe four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter,the Raiders had won by one point. Neither team scored more than 100 points. What was thetotal number of points scored by the two teams in the first half?

(A) 30 (B) 31 (C) 32 (D) 33 (E) 34

25 Let a > 0, and let P (x) be a polynomial with integer coe"cients such that

P (1) = P (3) = P (5) = P (7) = a, and

P (2) = P (4) = P (6) = P (8) = " a.

What is the smallest possible value of a?

(A) 105 (B) 315 (C) 945 (D) 7! (E) 8!







USAAMC 12/AHSME

2010

A

1 What is (20! (2010! 201)) + (2010! (201! 20))?

(A) ! 4020 (B) 0 (C) 40 (D) 401 (E) 4020

2 A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip,which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1fewer than on the previous trip. How many tourists did the ferry take to the island that day?

(A) 585 (B) 594 (C) 672 (D) 679 (E) 694

3 Rectangle ABCD, pictured below, shares 50

A B

CD

E F

GH

(A) 4 (B) 5 (C) 6 (D) 8 (E) 10

4 If x < 0, then which of the following must be positive?

(A) x|x| (B) ! x2 (C) ! 2x (D) ! x!1 (E) 3

"x

5 Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shota bullseye scores 10 points, with other possible scores being 8, 4, 2, 0 points. Chelsea alwaysscores at least 4 points on each shot. If Chelsea’s next n shots are bulleyes she will beguaranteed victory. What is the minimum value for n?

(A) 38 (B) 40 (C) 42 (D) 44 (E) 46

6 A palindrome, such as 83438, is a number that remains the same when its digits are reversed.The numbers x and x + 32 are three-digit and four-digit palindromes, respectively. What isthe sum of the digits of x?

(A) 20 (B) 21 (C) 22 (D) 23 (E) 24

http://www.artofproblemsolving.com/This file was downloaded from the AoPS Math Olympiad Resources Page Page 1











USAAMC 12/AHSME

2010

7 Logan is constructing a scaled model of his town. The city’s water tower stands 40 metershigh, and the top portion is a sphere that holes 100, 000 liters of water. Logan’s miniaturewater tower holds 0.1 liters. How tall, in meters, should Logan make his tower?

(A) 0.04 (B) 0.4! (C) 0.4 (D) 4

! (E) 4

8 Triangle ABC has AB = 2 · AC. Let D and E be on AB and BC, respectively, such that!BAE = !ACD. Let F be the intersection of segments AE and CD, and suppose that!CFE is equilateral. What is !ACB?

(A) 60! (B) 75! (C) 90! (D) 105! (E) 120!

9 A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center ofeach face. The edges of each cut are parallel to the edges of the cube, and each hole goes allthe way through the cube. What is the volume, in cubic inches, of the remaining solid?

(A) 7 (B) 8 (C) 10 (D) 12 (E) 15

10 The first four terms of an arithmetic sequence are p, 9, 3p" q, and 3p+ q. What is the 2010th

term of the sequence?

(A) 8041 (B) 8043 (C) 8045 (D) 8047 (E) 8049

11 The solution of the equation 7x+7 = 8x can be expressed in the form x = logb 77. What is b?

(A) 715 (B) 7

8 (C) 87 (D) 15

8 (E) 157

12 In a magical swamp there are two species of talking amphibians: toads, whose statements arealways true, and frogs, whose statements are always false. Four amphibians, Brian, Chris,LeRoy, and Mike live together in the swamp, and they make the following statements:

Brian: ”Mike and I are di!erent species.” Chris: ”LeRoy is a frog.” LeRoy: ”Chris is a frog.”Mike: ”Of the four of us, at least two are toads.”

How many of these amphibians are frogs?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

13 For how many integer values of k do the graphs of x2 + y2 = k2 and xy = k not intersect?

(A) 0 (B) 1 (C) 2 (D) 4 (E) 8

14 Nondegenerate !ABC has integer side lengths, BD is an angle bisector, AD = 3, andDC = 8. What is the smallest possible value of the perimeter?

(A) 30 (B) 33 (C) 35 (D) 36 (E) 37














USAAMC 12/AHSME

2010

15 A coin is altered so that the probability that it lands on heads is less than 12 and when the

coin is flipped four times, the probability of an equal number of heads and tails is 16 . What

is the probability that the coin lands on heads?

(A)!

15"36 (B) 6"

!6!

6+212 (C)

!2"12 (D) 3"

!3

6 (E)!

3"12

16 Bernardo randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and arrangesthem in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbersfrom the set {1, 2, 3, 4, 5, 6, 7, 8} and also arranges them in descending order to form a 3-digitnumber. What is the probability that Bernardo’s number is larger than Silvia’s number?

(A) 4772 (B) 37

56 (C) 23 (D) 49

72 (E) 3956

17 Equiangular hexagon ABCDEF has side lengths AB = CD = EF = 1 and BC = DE =FA = r. The area of "ACE is 70

(A) 4!

33 (B) 10

3 (C) 4 (D) 174 (E) 6

18 A 16-step path is to go from (#4,#4) to (4, 4) with each step increasing either the x-coordinateor the y-coordinate by 1. How many such paths stay outside or on the boundary of the square#2 $ x $ 2, #2 $ y $ 2 at each step?

(A) 92 (B) 144 (C) 1568 (D) 1698 (E) 12,800

19 Each of 2010 boxes in a line contains a single red marble, and for 1 $ k $ 2010, the box in thekth position also contains k white marbles. Isabella begins at the first box and successivelydraws a single marble at random from each box, in order. She stops when she first draws ared marble. Let P (n) be the probability that Isabella stops after drawing exactly n marbles.What is the smallest value of n for which P (n) < 1

2010?

(A) 45 (B) 63 (C) 64 (D) 201 (E) 1005

20 Arithmetic sequences (an) and (bn) have integer terms with a1 = b1 = 1 < a2 $ b2 andanbn = 2010 for some n. What is the largest possible value of n?

(A) 2 (B) 3 (C) 8 (D) 288 (E) 2009

21 The graph of y = x6 # 10x5 + 29x4 # 4x3 + ax2 lies above the line y = bx + c except at threevalues of x, where the graph and the line intersect. What is the largest of those values?

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

22 What is the minimum value of f(x) = |x# 1| + |2x# 1| + |3x# 1| + · · · + |119x# 1|?(A) 49 (B) 50 (C) 51 (D) 52 (E) 53

23 The number obtained from the last two nonzero digits of 90! is equal to n. What is n?

(A) 12 (B) 32 (C) 48 (D) 52 (E) 68















USAAMC 12/AHSME

2010

24 Let f(x) = log10(sin(!x) · sin(2!x) · sin(3!x) · · · sin(8!x)). The intersection of the domain off(x) with the interval [0, 1] is a union of n disjoint open intervals. What is n?

(A) 2 (B) 12 (C) 18 (D) 22 (E) 36

25 Two quadrilaterals are considered the same if one can be obtained from the other by a rotationand a translation. How many di!erent convex cyclic quadrilaterals are there with integer sidesand perimeter equal to 32?

(A) 560 (B) 564 (C) 568 (D) 1498 (E) 2255








USAAMC 12/AHSME

2010

B

1 Makayla attended two meetings during her 9-hour work day. The first meeting took 45minutes and the second meeting took twice as long. What percent of her work day was spentattending meetings?

(A) 15 (B) 20 (C) 25 (D) 30 (E) 35

2 A big L is formed as shown. What is its area?

5

2

2

8

(A) 22 (B) 24 (C) 26 (D) 28 (E) 30

3 A ticket to a school play costs x dollars, where x is a whole number. A group of 9th gradersbuys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of$64. How many values of x are possible?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

4 A month with 31 days has the same number of Mondays and Wednesdays. How many of theseven days of the week could be the first day of this month?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

5 Lucky Larry’s teacher asked him to substitute numbers for a, b, c, d, and e in the expressiona ! (b ! (c ! (d + e))) and evaluate the result. Larry ignored the parentheses but addedand subtracted correctly and obtained the correct result by coincedence. The numbers Larrysubstituted for a, b, c, and d were 1, 2, 3, and 4, respectively. What number did Larrysubstitute for e?

(A) ! 5 (B) ! 3 (C) 0 (D) 3 (E) 5











USAAMC 12/AHSME

2010

6 At the beginning of the school year, 50% of all students in Mr. Well’s math class answered”Yes” to the question ”Do you love math”, and 50% answered ”No.” At the end of the schoolyear, 70% answered ”Yes” and 30% answered ”No.” Altogether, x% of the students gave adi!erent answer at the beginning and end of the school year. What is the di!erence betweenthe maximum and the minimum possible values of x?

(A) 0 (B) 20 (C) 40 (D) 60 (E) 80

7 Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 milesper hour if it is raining. Today she drove in the sun in the morning and in the rain in theevening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?

(A) 18 (B) 21 (C) 24 (D) 27 (E) 30

8 Every high school in the city of Euclid sent a team of 3 students to a math contest. Eachparticipant in the contest received a di!erent score. Andrea’s score was the median among allstudents, and hers was the highest score on her team. Andrea’s teammates Beth and Carlaplaced 37th and 64th, respectively. How many schools are in the city?

(A) 22 (B) 23 (C) 24 (D) 25 (E) 26

9 Let n be the smallest positive integer such that n is divisible by 20, n2 is a perfect cube, andn3 is a perfect square. What is the number of digits of n?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

10 The average of the numbers 1, 2, 3, ..., 98, 99, and x is 100x. What is x?

(A) 49101 (B) 50

101 (C) 12 (D) 51

101 (E) 5099

11 A palindrome between 1000 and 10, 000 is chosen at random. What is the probability that itis divisible by 7?

(A)110

(B)19

(C)17

(D)16

(E)15

12 For what value of x does

log!2

!x + log2 x + log4(x

2) + log8(x3) + log16(x

4) = 40?

(A) 8 (B) 16 (C) 32 (D) 256 (E) 1024

13 In "ABC, cos(2A#B) + sin(A + B) = 2 and AB = 4. What is BC?

(A)!

2 (B)!

3 (C) 2 (D) 2!

2 (E) 2!

3

14 Let a, b, c, d, and e be positive integers with a + b + c + d + e = 2010, and let M be thelargest of the sums a + b, b + c, c + d, and d + e. What is the smallest possible value of M?

(A) 670 (B) 671 (C) 802 (D) 803 (E) 804















USAAMC 12/AHSME

2010

15 For how many ordered triples (x, y, z) of nonnegative integers less than 20 are there exactlytwo distinct elements in the set {ix, (1 + i)y, z}, where i =

!"1?

(A) 149 (B) 205 (C) 215 (D) 225 (E) 235

16 Positive integers a, b, and c are randomly and independently selected with replacement fromthe set {1, 2, 3, . . . , 2010}. What is the probability that abc + ab + a is divisible by 3?

(A)13

(B)2981

(C)3181

(D)1127

(E)1327

17 The entries in a 3 # 3 array include all the digits from 1 through 9, arranged so that theentries in every row and column are in increasing order. How many such arrays are there?

(A) 18 (B) 24 (C) 36 (D) 42 (E) 60

18 A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosenindependently and at random. What is the probability the the frog’s final position is no morethan 1 meter from its starting position?

(A) 16 (B) 1

5 (C) 14 (D) 1

3 (E) 12

19 A high school basketball game between the Raiders and Wildcats was tied at the end of thefirst quarter. The number of points scored by the Raiders in each of the four quarters formedan increasing geometric sequence, and the number of points scored by the Wildcats in each ofthe four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter,the Raiders had won by one point. Neither team scored more than 100 points. What was thetotal number of points scored by the two teams in the first half?

(A) 30 (B) 31 (C) 32 (D) 33 (E) 34

20 A geometric sequence (an) has a1 = sinx, a2 = cos x, and a3 = tanx for some real numberx. For what value of n does an = 1 + cos x?

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

21 Let a > 0, and let P (x) be a polynomial with integer coe!cients such that

P (1) = P (3) = P (5) = P (7) = a, and

P (2) = P (4) = P (6) = P (8) = " a.

What is the smallest possible value of a?

(A) 105 (B) 315 (C) 945 (D) 7! (E) 8!

22 Let ABCD be a cyclic quadrilateral. The side lengths of ABCD are distinct integers lessthan 15 such that BC · CD = AB · DA. What is the largest possible value of BD?

(A)!

3252 (B)

!185 (C)

!3892 (D)

!4252 (E)

!5332














USAAMC 12/AHSME

2010

23 Monic quadratic polynomials P (x) and Q(x) have the property that P (Q(x)) has zeroes atx = !23,!21,!17, and!15, and Q(P (x)) has zeroes at x = !59,!57,!51, and!49. Whatis the sum of the minimum values of P (x) and Q(x)?

(A) -100 (B) -82 (C) -73 (D) -64 (E) 0

24 The set of real numbers x for which

1x! 2009

+1

x! 2010+

1x! 2011

" 1

is the union of intervals of the form a < x # b. What is the sum of the lengths of theseintervals?

(A) 1003335 (B) 1004

335 (C) 3 (D) 403134 (E) 202

67

25 For every integer n " 2, let pow(n) be the largest power of the largest prime that divides n.For example pow(144) = pow(24 · 32) = 32. What is the largest integer m such that 2010m

divides5300!

n=2

pow(n)?

(A) 74 (B) 75 (C) 76 (D) 77 (E) 78









USAAIME

2010

I

1 Maya lists all the positive divisors of 20102. She then randomly selects two distinct divisorsfrom this list. Let p be the probability that exactly one of the selected divisors is a perfectsquare. The probability p can be expressed in the form m

n , where m and n are relativelyprime positive integers. Find m + n.

2 Find the remainder when 9! 99! 999! · · ·! 99 · · · 9! "# $999 9’s

is divided by 1000.

3 Suppose that y = 34x and xy = yx. The quantity x+ y can be expressed as a rational number

rs , where r and s are relatively prime positive integers. Find r + s.

4 Jackie and Phil have two fair coins and a third coin that comes up heads with probability 47 .

Jackie flips the three coins, and then Phil flips the three coins. Let mn be the probability that

Jackie gets the same number of heads as Phil, where m and n are relatively prime positiveintegers. Find m + n.

5 Positive integers a, b, c, and d satisfy a > b > c > d, a+b+c+d = 2010, and a2"b2+c2"d2 =2010. Find the number of possible values of a.

6 Let P (x) be a quadratic polynomial with real coe!cients satisfying x2 " 2x + 2 # P (x) #2x2 " 4x + 3 for all real numbers x, and suppose P (11) = 181. Find P (16).

7 Define an ordered triple (A,B, C) of sets to be minimally intersecting if |A$B| = |B $C| =|C$A| = 1 and A$B$C = %. For example, ({1, 2}, {2, 3}, {1, 3, 4}) is a minimally intersectingtriple. Let N be the number of minimally intersecting ordered triples of sets for which eachset is a subset of {1, 2, 3, 4, 5, 6, 7}. Find the remainder when N is divided by 1000.

Note: |S| represents the number of elements in the set S.

8 For a real number a, let &a' denominate the greatest integer less than or equal to a. Let Rdenote the region in the coordinate plane consisting of points (x, y) such that &x'2+&y'2 = 25.The region R is completely contained in a disk of radius r (a disk is the union of a circle andits interior). The minimum value of r can be written as

!mn , where m and n are integers and

m is not divisible by the square of any prime. Find m + n.

9 Let (a, b, c) be the real solution of the system of equations x3 " xyz = 2, y3 " xyz = 6,z3"xyz = 20. The greatest possible value of a3 +b3 +c3 can be written in the form m

n , wherem and n are relatively prime positive integers. Find m + n.















USAAIME

2010

10 Let N be the number of ways to write 2010 in the form 2010 = a3 ·103 +a2 ·102 +a1 ·10+a0,where the ai’s are integers, and 0 ! ai ! 99. An example of such a representation is1 · 103 + 3 · 102 + 67 · 101 + 40 · 100. Find N .

11 Let R be the region consisting of the set of points in the coordinate plane that satisfy both|8 " x| + y ! 10 and 3y " x # 15. When R is revolved around the line whose equation is3y"x = 15, the volume of the resulting solid is m!

n!

p , where m, n, and p are positive integers,m and n are relatively prime, and p is not divisible by the square of any prime. Find m+n+p.

12 Let M # 3 be an integer and let S = {3, 4, 5, . . . ,m}. Find the smallest value of m such thatfor every partition of S into two subsets, at least one of the subsets contains integers a, b,and c (not necessarily distinct) such that ab = c.

Note: a partition of S is a pair of sets A, B such that A $B = %, A &B = S.

13 Rectangle ABCD and a semicircle with diameter AB are coplanar and have nonoverlappinginteriors. Let R denote the region enclosed by the semicircle and the rectangle. Line ! meetsthe semicircle, segment AB, and segment CD at distinct points N , U , and T , respectively.Line ! divides region R into two regions with areas in the ratio 1 : 2. Suppose that AU = 84,AN = 126, and UB = 168. Then DA can be represented as m

'n, where m and n are positive

integers and n is not divisible by the square of any prime. Find m + n.

14 For each positive integer n, let f(n) =!100

k=1(log10(kn)). Find the largest value of n for whichf(n) ! 300.

Note: (x) is the greatest integer less than or equal to x.

15 In *ABC with AB = 12, BC = 13, and AC = 15, let M be a point on AC such that theincircles of *ABM and *BCM have equal radii. Let p and q be positive relatively primeintegers such that AM

CM = pq . Find p + q.












USAAIME

2010

II

1 Let N be the greatest integer multiple of 36 all of whose digits are even and no two of whosedigits are the same. Find the remainder when N is divided by 1000.

2 A point P is chosen at random in the interior of a unit square S. Let d(P ) denote the distancefrom P to the closest side of S. The probability that 1

5 ! d(P ) ! 13 is equal to m

n , where mand n are relatively prime positive integers. Find m + n.

3 Let K be the product of all factors (b"a) (not necessarily distinct) where a and b are integerssatisfying 1 ! a < b ! 20. Find the greatest positive integer n such that 2n divides K.

4 Dave arrives at an airport which has twelve gates arranged in a straight line with exactly 100feet between adjacent gates. His departure gate is assigned at random. After waiting at thatgate, Dave is told the departure gate has been changed to a di!erent gate, again at random.Let the probability that Dave walks 400 feet or less to the new gate be a fraction m

n , wherem and n are relatively prime positive integers. Find m + n.

5 Positive numbers x, y, and z satisfy xyz = 1081 and (log10 x)(log10 yz) + (log10 y)(log10 z) =468. Find

!(log10 x)2 + (log10 y)2 + (log10 z)2.

6 Find the smallest positive integer n with the property that the polynomial x4 " nx + 63 canbe written as a product of two nonconstant polynomials with integer coe"cients.

7 Let P (z) = z3 + az2 + bz + c, where a, b, and c are real. There exists a complex numberw such that the three roots of P (z) are w + 3i, w + 9i, and 2w " 4, where i2 = " 1. Find|a + b + c|.

8 Let N be the number of ordered pairs of nonempty sets A and B that have the followingproperties:

A # B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, A $ B = %, The number of elements of A is notan element of A, The number of elements of B is not an element of B.

Find N .

9 Let ABCDEF be a regular hexagon. Let G, H, I, J , K, and L be the midpoints of sidesAB, BC, CD, DE, EF , and AF , respectively. The segments AH, BI, CJ , DK, EL, andFG bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to thearea of ABCDEF be expressed as a fraction m

n where m and n are relatively prime positiveintegers. Find m + n.















USAAIME

2010

10 Find the number of second-degree polynomials f(x) with integer coe!cients and integer zerosfor which f(0) = 2010.

11 Define a T-grid to be a 3! 3 matrix which satisfies the following two properties:

(1) Exactly five of the entries are 1’s, and the remaining four entries are 0’s. (2) Among theeight rows, columns, and long diagonals (the long diagonals are {a13, a22, a31} and {a11, a22, a33},no more than one of the eight has all three entries equal.

Find the number of distinct T-grids.

12 Two noncongruent integer-sided isosceles triangles have the same perimeter and the samearea. The ratio of the lengths of the bases of the two triangles is 8 : 7. Find the minimumpossible value of their common perimeter.

13 The 52 cards in a deck are numbered 1, 2, . . . , 52. Alex, Blair, Corey, and Dylan each picks acard from the deck without replacement and with each card being equally likely to be picked,The two persons with lower numbered cards from a team, and the two persons with highernumbered cards form another team. Let p(a) be the probability that Alex and Dylan areon the same team, given that Alex picks one of the cards a and a + 9, and Dylan picks theother of these two cards. The minimum value of p(a) for which p(a) " 1

2 can be written asmn . where m and n are relatively prime positive integers. Find m + n.

14 In right triangle ABC with right angle at C, !BAC < 45 degrees and AB = 4. Point P onAB is chosen such that !APC = 2!ACP and CP = 1. The ratio AP

BP can be representedin the form p + q

#r, where p, q, r are positive integers and r is not divisible by the square of

any prime. Find p + q + r.

15 In triangle ABC, AC = 13, BC = 14, and AB = 15. Points M and D lie on AC withAM = MC and !ABD = !DBC. Points N and E lie on AB with AN = NB and!ACE = !ECB. Let P be the point, other than A, of intersection of the circumcircles of$AMN and $ADE. Ray AP meets BC at Q. The ratio BQ

CQ can be written in the form mn ,

where m and n are relatively prime positive integers. Find m% n.












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